Mathematics - 2012-09-20 (page 1 of 3)

@robjohn As I was saying before, I have a function: y = h(s(a(f)), a(f)). I want to express how a change in f affects y. I'm not sure how to express this with regards to total derivs and partials.

robjohn

@JG hmm, the align environment implies displaystyle

J G

@robjohn I know the expression should be something like: dy/df = dh/ds*ds/da*da/df + dh/da*da/df

robjohn

@JG just a sec

J G

@robjohn I'm not sure if that should be partials or total derivs. Like partial y/ partial f. on the left hand side. Of say partial h/partial s and so forth on the right hand side.

robjohn

3:10 AM

$\displaystyle\frac{\mathrm{d}y}{\mathrm{d}f}=h_1(s(a(f)),a(f))s'(a(f))a'(f) +h_2(s(a(f)),a(f))a'(f)$

where $h_1$ is the partial of $h$ wrt the first variable and $h_2$ is the partial of $h$ wrt the second variable.

J G

@robjohn so that is dy / df as in total deriv. all of the others are partial derivs? or is just the h a partial deriv? Is it dy / df = partial h / partial s * ds / da * da / df . . . or is it partial h / partial s * partial s / partial a * partial a / partial f . . . .

robjohn

Since $s$ and $a$ only depend on one variable, there is no need for a partial.

J G

@robjohn so just partial h? all others total?

robjohn

@JG yep

J G

@robjohn is it bad form to write out dy/df = partial h/partial s*ds/da*da/df + partial h/partial a*da/df

@robjohn in constrast to the way you wrote h_1 and h_2 ?

robjohn

3:18 AM

@JG No, that is fine, you just want to make sure the reader knows which argument is being derived.

J G

@robjohn Thank you!! May I ask another question? What do you mean by which argument is being defined?

robjohn

if "derived" is an actual verb in this situation.

J G

@robjohn What would (dy / df) / (ds / df) be?

anon

derived, not defined

J G

@robjohn oh, what do you mean by derived?

@robjohn i guess i mean argument, what do you mean ?

robjohn

3:20 AM

@JG whether you are taking the derivative with respect to the first or second argument.

@JG $h(x,y)$ --- $x$ is the first argument and $y$ is the second.

@JG argument = parameter

J G

@robjohn you mean in terms of the partial h / partial something?

robjohn

@JG yes

J G

@robjohn how is that not clear in the way that i am doing it now? i guess I don't see what you are recommending instead?

robjohn

@JG well, first of all, you are not taking the derivative of $h$ wrt $s$ or $a$, so the notation might be confusing to some, but looking at the formula, one can figure out what you mean. If $s(a(f))$ and $a(f)$ were not the arguments, then it would be even less clear.

For example, how would you write $\frac{\partial}{\partial x}f(x+y,xy)$

J G

@robjohn so I should write displaystyle\frac{\mathrm{d}y}{\mathrm{d}f}=h_1(s(a(f)),a(f))s'(a(f))a'(f) +h_2(s(a(f)),a(f))a'(f) and how should h_1 and h_2 be defined? h_1 = ? h_2 = ?

@robjohn yes, your example poses the same conundrum.

robjohn

3:29 AM

I usually write something like $\partial_1$ and $\partial_2$, etc for partials wrt argument positions.

J G

@robjohn so do i need to define h_1 and h_2 or are they obvious?

robjohn

In which case we would have $\frac{\partial}{\partial x}f(x+y,xy)=\partial_1f(x+y,xy)+y\partial_2f(x+y,xy)$

@JG you should probably say what they are.

J G

@robjohn so what are they exactly? that is what i'm confused by. h_1 = ?

robjohn

@JG $h_1(x,y)=\lim\limits_{d\to0}\frac{h(x+d,y)-h(x,y)}{d}$

@JG $h_2(x,y)=\lim\limits_{d\to0}\frac{h(x,y+d)-h(x,y)}{d}$

J G

@robjohn ok, that makes sense!

@robjohn and what would (dy / df) / (ds / df) be?

robjohn

3:37 AM

@JG I don't know what that would represent...

J G

@robjohn I want to say what dy / ds would be . . .

robjohn

@JG it would be the relative change of $y$ with respect to $s$

J G

@robjohn it is the case that \frac{dy}{ds} = \frac{\frac{dy}{df}}\frac{ds}{df}} right?

robjohn

@JG do you have the ChatJax bookmark installed?

J G

@robjohn I don't know what that means.

John Junior

3:39 AM

math.ucla.edu/~robjohn/math/mathjax.html

robjohn

@JG Do you see the Latex rendered or simply as text?

J G

@robjohn text

@robjohn latex syntax i should say

robjohn

@JG Oh!!! You need to install the bookmark in your bookmark bar

J G

@robjohn i see that link by john. which do i download?

robjohn

math.ucla.edu/~robjohn/math/mathjax.html

@JG You should at least install the ChatJax one

"start ChatJax"

"render MathJax" is nice for use on pages where the MathJax does not render automatically.

J G

3:43 AM

@robjohn ok, i installed it.

robjohn

@JG run it

anon

now see if it works - $\LaTeX$

robjohn

9 mins ago, by robjohn

@JG $h_1(x,y)=\lim\limits_{d\to0}\frac{h(x+d,y)-h(x,y)}{d}$

J G

$\frac{dy}{ds}$

ok, it works!

robjohn

If all this looks rendered you are good to go

J G

3:45 AM

so what is $\frac{dy}{ds}$ ?

$\frac{\frac{dy}{df}}{\frac{ds}{df}}$

robjohn

@JG well if you parametrically plot $y$ vs $s$ it would be the slope of that curve

J G

@robjohn sorry, i'm not fully following. what would that be analyticall?

robjohn

@JG It is just the ratio of the rates of change of $y$ vs the rate of change of $s$

There is only one independent variable, $f$

J G

@robjohn i mean in terms of h

robjohn

@JG I don't know what it would mean...

J G

3:51 AM

like i wrote $\frac{dy}{df}$ in terms of $f$, $a$, $h$, and $s$

can $\frac{dy}{ds}$ be expressed in terms of $f$, $a$, and $h$?

robjohn

@JG the $s$ that is in $y$ has $a(f)$ as its argument, but $\frac{\mathrm{d}s(f)}{\mathrm{d}f}$ it it doesn't

J G

@robjohn then what is $\frac{dy}{ds}$

@robjohn and what if $y = h(s(f), a)$. would $\frac{dy}{ds}$ differ?

robjohn

@JG $s$ is not an independent variable, and depending on the domains and ranges of things I don't even know if $y$ and $s$ can have the same argument. This is not an easy question

@JG does $a$ depend on $f$?

Gustavo Bandeira

0

Q: Show that the tractrix is orthogonal to...

Show that the tractrix discussed in Example 1.17 is orthogonal to the lower half of each circle with radius a and center on the positive y-axis. Example 1.17: [y']^2=x^2y'' We note that y is missing, so we make the substitution p=y', p'=y''. Thus the equation becomes: p^2=x^2p' now use sep...

I've edited this.

J G

@robjohn well that is why i had asked about dividing $\frac{dy}{df} / \frac{ds}{df}$

@robjohn in the 2nd example, no.

@robjohn we know $\frac{dy}{df}$ and $\frac{ds}{df} = \frac{ds}{da} * \frac{da}{df}$ ? is that right?

robjohn

3:59 AM

@JG and that is simply a quotient of rates. However since we only see $s(a(f))$ I don't know if $s(f)$ even makes sense.

@JG $a$ is not an independent variable. It is a function of $f$

J G

@robjohn I'm not seeing what would $\frac{dy}{ds}$ be then analytically . . .

robjohn

we could talk about $\frac{\mathrm{d}s(a(f))}{\mathrm{d}f}$

@JG I don't either

@JG if $a$ maps from the domain of $s$ to the domain of $s$ then $s(f)$ makes sense

@JG Is your teacher asking you what that means?

J G

@robjohn no this is not a problem set or anything.

@robjohn should i explain in words?

robjohn

@JG sure

J G

@robjohn y is wage. s is years of schooling. a is IQ. imagine f is # of cigs a mother smoked while she as pregnant with the person.

@robjohn wages higher if schooling is higher.

@robjohn wages higher if IQ is higher.

@robjohn schooling is also higher if IQ is higher. that is, IQ has a direct effect on wages and a direct effect on schooling.

@robjohn number of cigs reduces IQ.

@robjohn h is a function that is increasing in s and a. second derivs are negative. cross derivative is positive.

math101

4:08 AM

Any one familiar with numerical analysis

robjohn

So the formula assumes that all other parameters take some predetermined value and all we are changing is the cigarette consumption

J G

@robjohn yes.

@math101 vaguely.

@robjohn everything is pinned down by f, we can assume.

robjohn

@JG So when you are asking about $s$ you are asking about $s(a(f))$ only

J G

@robjohn the first thing i want to show is that f affects y through 2 channels: the first is directly through IQ (a). the second is indirectly via schooling.

@robjohn we already showed that with $\frac{dy}{df}$

@robjohn i think. i'm not fully sure what the distinction implies.

robjohn

So when you say $\frac{\mathrm{d}s}{\mathrm{d}f}$ you mean $\frac{\mathrm{d}s(a(f))}{\mathrm{d}f}$

J G

4:11 AM

@robjohn the second thing i want to show is $\frac{dy}{ds}$

@robjohn yes i guess.

@robjohn it is like when i say $\frac{dy}{df}$ that is $\frac{d h(s(a(f)), a(f))}{df}$

@robjohn or i think that is the same as $\frac{dy}{df}$ is equivalent to $\frac{d(s, a; f}{df}$

robjohn

@JG well, there are two ways you might mean that, you might mean $\partial_1h(s(a(f)),a(f))$

J G

@robjohn the first thing i showed was if cigs declines, how much would y increase?

@robjohn now i want to say, the increase in schooling due to a decline in cigs, what is the corresponding increase in y look like?

robjohn

@JG the other thing might be $\frac{ h_1(s(a(f)),a(f))s'(a(f))a'(f) +h_2(s(a(f)),a(f))a'(f)}{s'(a(f))a'(f) }$

J G

@robjohn i'm not following that expression

@robjohn yes, that is kind of what i want to express. is there a way to simplify that?

robjohn

@JG no

J G

4:19 AM

@robjohn like i cannot divide through by $a'(f)$?

robjohn

@JG well, yes you can do that.

J G

@robjohn i just mean what can be done to make that fraction look as simple as possible?

robjohn

@JG note that $f$ is only seen as an argument to $a$ so you can lift everything to a function of $a$

J G

can i write that as something like $h_1 + \frac{h_2}{s'(a(f))}$

robjohn

$y=h(s(a)),a)$

J G

4:22 AM

@robjohn yes, i agree. so then what would be the simplest way to express that fraction?

robjohn

$\frac{\mathrm{d}y}{\mathrm{d}s}=\frac{h_1(s(a),a)s'(a)+h_2(s(a),a)}{s'(a)} =h_1(s(a),a)+\frac{h_2(s(a),a)}{s'(a)}$

J G

@robjohn can i separate that fraction and divide the first one through by $s'(a)$?

@robjohn yes! i agree! that is what i meant.

@robjohn can i ask you about how to interpret that right-hand side? does that seem to have any interpretation?

robjohn

@JG none that I can think of right off.

J G

@robjohn i mainly want to make the point that suppose we had $y = h(s(f), a)$ that in $\frac{dy}{ds}$ the second term wouldn't be there, right?

robjohn

@JG You have to be careful about what is dependent on what. I am not sure what the independent variables are there.

J G

4:31 AM

@robjohn same situation but cigs only affects schooling and not IQ.

@robjohn i want to distinguish that in that setting there is only one channel in $\frac{dy}{ds}$, the $h_1 (s, a)$

robjohn

@JG well then $s$ and $a$ are independent and it is not certain what $\frac{\mathrm{d}y}{\mathrm{d}s}$ means

@JG the partial derivative would make sense

J G

@robjohn if cigs decrease, how would the corresponding increase in schooling affect the corresponding increase in wages?

robjohn

@JG but since $y$ is dependent on two independent variables, the total derivative doesn't make sense.

leo

@robjohn, you provide a different approach to the one in the book, thanks!

robjohn

@leo how do they do it?

@leo Or are you talking about the way I did it in chat?

J G

4:38 AM

@robjohn let's put another way. in the original function, i want to make clear that there are 2 channels. a direct one via IQ and an indirect one from IQ -> schooling -> wages.

robjohn

@JG sure, that is fine, the way you originally defined $y$

J G

@robjohn is it possible to quantify the share of the change in y that is due to the indirect effect via schooling? would that be $\frac{ds}{df} / \frac{dy}{df}$ ?

robjohn

if you want to attribute the rate of change to different effects, I would put $h_1(s(a),a)s'(a)$ as the rate of change due to the schooling and $h_2(s(a),a)$ as the effect of IQ

Or each times $a'$ if you want it in terms of $f$

leo

@robjohn he does it in yet another way. I'll add it as another answer. It deserves it

robjohn

@leo I would be interested. I thought I was following the suggestions in the assignment.

J G

4:50 AM

@robjohn yes, i want $f$ so $h_1(.) s'(a)a'(f) + h_2(.)a'(f)$ ?

robjohn

@JG That's what I would say. Add them up and you get the total rate

John Junior

@waiwai933 Hi, welcome!

J G

@robjohn is it possible to quantify the share of the change in y that is due to the indirect effect via schooling? would that be dsdf/dydf ?

robjohn

@JG that would be the total change in $y$ not the partial change

wj32

can anyone quickly think of a linear map with spectrum {0} that is NOT nilpotent?

J G

4:57 AM

@robjohn what do you mean? can i say that the first term in the sum is the complete indirect effect via schooling and the second is the direct effect via IQ?

wj32

i can think of a very convoluted example

J G

@robjohn therefore the total effect should be x% direct effect and (100-x)% effect via schooling?

robjohn

@wj32 It would have to be infinite dimensional I think

John Junior

The indirect effect via schooling comes when you forget everything you learned in school :)

wj32

@robjohn right, but the "simplest" example i can think of is this:

robjohn

4:58 AM

@JG That is the idea of the partial derivatives.

J G

@robjohn i thought the partials only came into play with the h terms? i'm not sure i see your point.

wj32

Let $V$ be the set of all functions $f:\mathbb{N}\rightarrow\mathbb{R}$ such that $\lim_{k \rightarrow \infty} f(k)=0$. Define $\tau:V \rightarrow V$ by $$(\tau f)(k)=\frac{f(k+1)}{k+1}$$

robjohn

@JG $y$ is $h$, but seen solely as a function of $f$

wj32

@robjohn do you happen to have any good examples?

J G

@robjohn i'm not following. is the thing i said about the x% not correct?

robjohn

5:03 AM

@wj32 I don't know if this works, but how about shifting a sequence and replacing the starting term with 0.

J G

@robjohn is there another way to express h_1 without directly invoking the limit definition?

robjohn

@JG That is the definition of the partial derivative.

J G

@robjohn i just mean is there another way to write that without using the limit?

@robjohn or even a brief say to put say it in words?

wj32

@robjohn thanks but i don't think it works

robjohn

@JG I'm not exactly sure what you mean... how about using $\partial$s?

wj32

5:11 AM

if you shift right by 1 place and use 0 for the new first value, 0 is not an eigenvalue

J G

@robjohn what do you mean exactly?

robjohn

@JG partial derivative with respect to the first argument.

wj32

if you shift left by 1 place and replace the first value with 0, we have an eigenvalue of 1 with eigenvector [0,1,1,1,1,...]

J G

@robjohn you mean $h_1 = \frac{\partial h}{\partial s}$ ?

wj32

i guess i'll just have to stick with the convoluted example

robjohn

5:14 AM

@wj32 no that is not an eigenvector

@wj32 that would map to $\{0,0,1,1,1,1,\dots\}$

J G

@robjohn and how do i turn off the latex typing , the Jax? I want to see the latex code you typed up to write the derivatives

wj32

@robjohn so you're shifting right?

robjohn

@JG right click on the rendered LaTeX and there will be options to show the code

@wj32 yes

wj32

@robjohn but then 0 is not an eigenvalue

@robjohn no information is lost by shifting right...

J G

What does the $\displaystyle$ do?

robjohn

5:16 AM

@JG you can also refresh your browser and it will stop the rendering. You just need to run the bookmark again.

@wj32 what would be the spectrum of that operator?

J G

so can i write? \begin{flalign}
\frac{\mathrm{d}\ln(y)}{\mathrm{d}f} = \frac{\partial h}{\partial a} a'(f) + \frac{\partial h}{\partial s} s'(a(f))a'(f)
\end{flalign}

robjohn

@JG use align, not flalign

J G

$
\frac{\mathrm{d}\ln(y)}{\mathrm{d}f} = \frac{\partial h}{\partial a} a'(f) + \frac{\partial h}{\partial s} s'(a(f))a'(f)$

@robjohn $\frac{\mathrm{d}\ln(y)}{\mathrm{d}f} = \frac{\partial h}{\partial a} a'(f) + \frac{\partial h}{\partial s} s'(a(f))a'(f)$

$\frac{dy}{df}$

wj32

@robjohn i'm guessing that it's empty, but i'm looking for operators with 0 as the only eigenvalue

J G

@robjohn is there anything else latex or mathematicall? like \mathrm{d} ?

@robjohn are the partials ok or is h_1 and h_2 instead needed?

robjohn

5:23 AM

Spectrum (functional analysis)

In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an ...

@wj32 That says that $0$ is in the spectrum, but you want a $0$ eigenvalue.

wj32

@robjohn sorry, i should have clarified my definitions beforehand

@robjohn i'm using eigenvalue = $\lambda$ with $\tau v = \lambda v$ for some nonzero $v$ and spectrum = set of all eigenvalues

@robjohn from Advanced Linear Algebra (Roman)

robjohn

@JG You can define \d in a latex document, but definitions in chat only last as long as it stays in the scrollback and is bad because it can mess up everyone on the chat

@wj32 Yes, I understand now :-)

J G

@robjohn what do you mean exactly?

robjohn

@JG you can use \newcommand to define \d to be \mathrm{d}

J G

@robjohn oh ok. and you recommend doing that?

robjohn

5:33 AM

but NOT in chat.

J G

@robjohn is there anything else that you would recommend in writing that statement?

robjohn

It will define that for everyone in chat for an unspecified time, and if you mess up, it messes up everyone for an unspecified time

J G

@robjohn as i said, i am wondering whether there is another way to write that without needing to write out limit definitions.

robjohn

So we frown on using \newcommand in chat

J G

@robjohn ok, i'm not doing it. you have my word!

robjohn

5:35 AM

@JG any of the ways you write a partial derivative.

J G

@robjohn what do you mean exactly?

robjohn

@JG well, I am not quite sure what you are asking. It sounds as if you are asking how to write a partial derivative without using the limit definition

J G

@robjohn i meant is it wrong to write $\frac{\mathrm{d}\ln(y)}{\mathrm{d}f} = \frac{\partial h}{\partial a} a'(f) + \frac{\partial h}{\partial s} s'(a(f))a'(f)$

$\frac{d\ln(y)}{df} = \frac{\partial h}{\partial a} a'(f) + \frac{\partial h}{\partial s} s'(a(f))a'(f)$

robjohn

@JG other than the $\ln(y)$ it looks okay, but the $\frac{\partial h}{\partial a}$ and $\frac{\partial h}{\partial s}$ might be misleading since they are partials with respect to the first and second argument.

J G

@robjohn you recommend using h_1 and h_2? is it obvious what that is? if not, do i have to use limits to define them? Sorry for all of the questions. i'm just triyng to understanding precisely.

robjohn

5:42 AM

@JG No, it would be nice to say that $h_1$ is the partial with respect to the first argument.

J G

@robjohn how can that be said?

robjohn

Or sometimes I use $\partial_1$

@JG just as I said it: $h_1$ is the partial of $h$ with respect to its first argument.

Jayesh Badwaik

@Evan This may be of help. It may not be good enough. If you can specify your requirement of why you want MATLAB along with C++, then I might be able to help you a little more.

@Evan If you are okay with c++ being your primary workhorse, then this kind of hacks will serve you well. If you want MATLAB to be your primary workhorse, then I am not so sure, and my guess is you might be better of using some sort of hack to transfer data using CSV files or something similar .

J G

@robjohn $\frac{dy}{df} = h_1 s'a(f) a'(f) + h_2 a'(f)$ where $h_1$ is partial of $h$ with respect to first argument ?

robjohn

@waiwai933: what does the IRIDIRID.. around the B mean?

Jayesh Badwaik

5:47 AM

@WillHunting 501! Finally.

robjohn

@JG and the second, too. You could say that $h_i$ is the partial derivative of $h$ with respect to its $i^{\mathrm{th}}$ argument.

@JayeshBadwaik Congratulations!

@JayeshBadwaik Now you're 511.

afk bbiab

Jayesh Badwaik

@robjohn Thank you. :-)

J G

@robjohn thank you! looks beautiful

hhh

6:24 AM

https://dl.dropbox.com/u/96742826/Shared_algGeo_1.pdf

<-- could someone help me to parametrise a hyperbola?

My assignment says that I should somehow see it when I draw lines there, pages 7-10.

Assignment given on pages 11-12, and I am wondering the last exercises about parametrization.

Deadline was yeasterday and I have handled the problems but I would still like to understand the parametrization, one professor said that look at the picture once in a similar problem.

errr dead-easay

Meysam

6:42 AM

Hey guys, what are numbers like 15 called? I mean they have only two factors (excluding 1 and 15) => 3 and 5

Gustavo Bandeira

6:56 AM

oh, nvm

I guess they are composite numbers.

John Junior

odd

Gustavo Bandeira

@JohnJunior Having only two factors $\Leftrightarrow$ odd number?

John Junior

@GustavoBandeira No. I said 15 is an odd number.

Zhen Lin

7:27 AM

semiprime

user19161

@Meysam Check out "semiprime", term courtesy of Zhen Lin. He did not ping you so I did.

user19161

@JayeshBadwaik Congrats! Now on to 1000. This site is the blackhole of productivity!

John Junior

semi-unprime?

Jayesh Badwaik

I am trying to improve without sacrificing my productivity! :-)
Also, I have a doubt, will you say that

\begin{equation}
f(x) = tan\left(\frac{\pi x}{2}\right)
\end{equation}

is convex on $(0,1)$

Jonas Teuwen

$\tan$ not $tan$.

3

BenjaLim

7:36 AM

@ZhenLin Hey

Zhen Lin

hi

Gustavo Bandeira

There's a number below our names, what is this number?

user19161

@robjohn waiwai usually does not chat. But he is the most professional mod on SE despite his age.

user19161

@GustavoBandeira Total SE rep.

Jonas Teuwen

@GustavoBandeira It is the number of stupid jokes you made.

Gustavo Bandeira

7:38 AM

@JonasTeuwen xD

John Junior

@JonasTeuwen That explains a lot.

user19161

@JayeshBadwaik Why not? Doesn't it satisfy the definition of convexity?

Jayesh Badwaik

@JonasTeuwen yeah, typo. :-( Cannot edit it now, so leave it for now. Will try to improve with time.

Jonas Teuwen

The exception is me, for me is the number of awesomeness.

Gustavo Bandeira

@WillHunting Isn't my total rep on SE shown on my flair? If yes, they're quite differente values, look: stackexchange.com/users/flair/1279610.png

user19161

7:39 AM

@GustavoBandeira Flair only includes sites with 200 rep. Chat includes all sites.

Gustavo Bandeira

@WillHunting Oh,ok.

user19161

There is nothing to flair about the flair.

BenjaLim

@ZhenLin Ok I was mistaken earlier we only know that the torsion part injects into $\Bbb{Z}/n\Bbb{Z}$

Gustavo Bandeira

@JonasTeuwen Of course....

user19161

@JayeshBadwaik This is a weird question. Isn't it trivial?

Jayesh Badwaik

7:40 AM

@vesszabo We are talking about functions with real values, not including infinity. For such functions convexity implies continuity in the interior of the interval. At the endpoints $g_n$ is continuous by design. — LVK Sep 16 at 17:34

Here LVK is talking only about real values, not infinity, I am not sure why.

The first line of the question text does not say anything about infinity, hence, trying to explore all possibilities.

8

Q: Sequence of convex functions

If a sequence of convex functions $\{ f_n \}$ on [0,1] converges pointwise to a continuous function f, then is the convergence uniform? The questions is almost identical to this one, except that the functions are not assumed to be continuous on [0,1]. Now I know that convex functions are continu...

convergence convex-analysis

user19161

@JayeshBadwaik Did you mean $2/\pi$ there? Oops.

Jayesh Badwaik

@WillHunting No, that is correct, except for $\tan$ being $tan$.

robjohn

@WillHunting Thanks. I was just intrigued by the avatar.

user19161

@JayeshBadwaik Sorry I may have misthought. Does that cross any asymptote?

Jayesh Badwaik

@WillHunting No. My point is in that question, consider the function $f_{n}$ defined as follows:

user19161

7:44 AM

@JayeshBadwaik Oh I see that it does not cross over any asymptote. It is CERTAINLY convex in that domain.

Jayesh Badwaik

\begin{align}
f_{n}(x) &= \frac{1}{n} \tan\left( \frac{\pi x}{2}\right) & x \in (0,1) \\
&= 0 &x=0,1
\end{align}

user19161

@JayeshBadwaik Anyway infinity does not come into play in what you showed above.

user19161

@JayeshBadwaik Sure, but this function is no longer convex in the bigger domain.

Jayesh Badwaik

@WillHunting Ahh, now I see it.

user19161

@JayeshBadwaik Duh.

user19161

7:48 AM

@JayeshBadwaik Also, no need to define the value at 0 separately, it can be included in $[0,1)$.

Jayesh Badwaik

@WillHunting Yeah, that I realized later, but left it as it is. Anyway, So, $f_{n}$ must not approach infinity, else, the continuity will be lost on the bigger domain. Hence, we deal only in finite values, and then the proof by LVK holds.

@WillHunting On Unix and Linux SE,

"I'd like to learn emacs in depth. The problem is that they divide in 2 categories:

Basics (That C-x C-s saves files and C-x C-c exists emacs etc.)
Everything"

:P

user19161

@JayeshBadwaik Aww, Emacs looks too hard for me, same for Arch.

Jonas Teuwen

@WillHunting It is not, install it.

@JayeshBadwaik Takes a while, but you learn while using.

Jayesh Badwaik

@WillHunting It is not.

Jonas Teuwen

Usually that is after spending lots of time and thinking: "this should be possible to do easier!"

Jayesh Badwaik

7:54 AM

@JonasTeuwen Yeah, I know. Have got started with org mode now.

Jonas Teuwen

@JayeshBadwaik Whoah, cool! :-).

I am trying to hook it well to Mendeley.

Maybe you can also install Mendeley and tell me your ideas. Mendeley is kickass.

It is free but not completely open source, but still very nice.

user19161

@JonasTeuwen For your bibliography?

Jonas Teuwen

Yes.

user19161

I thought you don't read books!

Jonas Teuwen

Mendeley exports to .bib, but I don't want to export the bibliography.

user19161

7:56 AM

Your bibliography should have one item only: Jonas's mind.

Jonas Teuwen

If I include something it should make one with only those.

Jayesh Badwaik

@JonasTeuwen Ohh. I will install it, no problem with that. I used mendeley sometime back I guess, did not understand it fully then I suppose, since I thought it was very bare. I will give a try again now.

Jonas Teuwen

@WillHunting No, that is wrong. I do read books, but not completely to learn a subject. Reference!

@JayeshBadwaik It is very useful.

user19161

My undergrad paper had no references. I used only my own mind.

Jonas Teuwen

Might take a while before you fully catch on.

@WillHunting Yeah, sure, nice, but for papers they want references.

user19161

7:57 AM

@JonasTeuwen But of course, what I did was trivial!

Jonas Teuwen

And if I have a big database, I don't want to include that into the local files, right 8-).

user19161

@JonasTeuwen Ah, references if and only if you use them of course.

Jonas Teuwen

Yes, in Mendeley you would have to select them manually and then export or just export all of them at once.

That sucks, you see.

Jayesh Badwaik

Okay. Hmm.
@WillHunting My undergrad paper had 12 references for a 32 pages. :P It was the shortest thesis in my class.

Jonas Teuwen

You want auto.

user19161

7:58 AM

@JayeshBadwaik Mine is a few pages shorter than yours. It broke the all time record I think. I think most people wrote about 100 pages.

Jonas Teuwen

Hmm, let me find mine... (it sucks).

Jayesh Badwaik

@JonasTeuwen You always want auto.

Jonas Teuwen

Here is mine.

@JayeshBadwaik Yes, things that can easily be done auto should be done auto. :D.

Jayesh Badwaik

@WillHunting Mine was an engineering one, so there was a lot of description and stuff. I have observed that math thesis are considerably shorter.

user19161

@JayeshBadwaik I observed that most of the long ones are redundant words.

user19161

8:00 AM

Nowadays people equate long with good.

Jayesh Badwaik

@WillHunting In math you mean? Hmm, in engineering no. If I had been able to complete my last experiment completely, the thesis would have swelled to around 100 pages, while still retaining the extreme terseness of the current one.

Jonas Teuwen

In a university nearby it is not allowed to be longer than 15 pages.

user19161

@JayeshBadwaik It must be a big experiment.

user19161

@JonasTeuwen Otherwise, one gets a PhD!

Jonas Teuwen

@WillHunting No.

They see more pages as more shit.

user19161

8:02 AM

@JonasTeuwen Hehe, I just shit a few minutes ago.

Jayesh Badwaik

@WillHunting It was. Basically, what happened was I had developed the a different tool for that simulation, but the simulation just would not converge for non-trivial problems. No point in describing the algorithms used there if they are not non-trivial.

user19161

@JayeshBadwaik You should have whisky when this happens.

Jayesh Badwaik

@WillHunting Naah, no one else has succeeded on the same problem and the only issue was a lack of a better solver. In that context, I was pretty proud of my work. :-) (It is not so much as I lacked a better solver that I was not able to interface and run it properly over the grid I had)

@JonasTeuwen De Broglie's PhD thesis was 8 pages I heard. Haven't checked if its true yet.

Jonas Teuwen

What is this shit with "pages", "books" or whatever. Are you guys scientists or managers? 8-).

Jayesh Badwaik

Managers FTW!!! How did the topic even start? Yup, I started it. :P :P
And my FTW is not the normal sense here. :P :P

239 downloads are there on Mendeley AUR package, that is one of the highest I have seen. Well. Good.

Jonas Teuwen

8:10 AM

@JayeshBadwaik It has its own updater as well.

Jayesh Badwaik

@JonasTeuwen Okay.

Transcript for

$Mathematics$ Mathematics